10-Sept., 1998 EE421, Lecture 2 1

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Lecture 2: Continuation of Sampling

Butterworth FiltersThe frequency response magnitude for an Nth order Butterworth filter is of the form:

– The cutoff frequency is f0.

– The roll-off rate is 20N dB/decade.

N2

0

0

ff

1

HfH

)(

10-Sept., 1998 EE421, Lecture 2 2

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Butterworth Filters

2nd order Sallen-Key low-pass filter (can be cascaded to create 4th, 6th, 8th, … order filters)

C

C

R R(1-K)Rf

Rf

+-

RC2

1f0

++

- -

input output

10-Sept., 1998 EE421, Lecture 2 3

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Butterworth Filters

Example 2.1: Suppose H0 = 1 and you want to select f0 for 1 dB (or less) attenuation over the band from 0 to 4 kHz.

at f = 4 kHz. So,

or

1

ff

110

fH20dBnattenuatioN2

010

10

log

)(log)(

110f4 10

N2

0

.

N21

100

110

4f

.

in kHz

10-Sept., 1998 EE421, Lecture 2 4

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Ideal Reconstruction

signal spectrum

sampled-signal spectrumideal reconstruction filter

Reconstructed-signal spectrum

f

f

f

10-Sept., 1998 EE421, Lecture 2 5

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Ideal Reconstruction

The impulse response of an ideal reconstructor is a sinc function, therefore an ideal reconstructor uses sinc interpolation.

sampled signal Interpolation signals

Interpolated signal

10-Sept., 1998 EE421, Lecture 2 6

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Practical Reconstruction

The staircase reconstructor is the simplest and most widely used reconstructor in practice. Basically, it is a sample and hold.

sampled signal Interpolation signals

Interpolated signalThe impulse response for a

staircase reconstructor is a

rectangle. Therefore, its frequency response is a

sinc!

10-Sept., 1998 EE421, Lecture 2 7

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Staircase Reconstruction

signal spectrum

staircase reconstruction filter

Reconstructed-signal spectrum

f

f

f

sampled-signal spectrum

surviving replicas (images)

10-Sept., 1998 EE421, Lecture 2 8

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Staircase Reconstruction

The staircase reconstructor should be followed by a low-pass filter to remove the residual images.

Also, a staircase reconstructor suffers from nonuniform pass-band attenuation. This can be corrected with a digital equalization filter with a response equal to an inverse sinc function over the pass-band.